Some properties of lower level-sets of convolutions

In the present paper we prove a certain lemma about the structure of "lower level-sets of convolutions", which are sets of the form $\{x \in \Z_N : 1_A*1_A(x) \leq \gamma N\}$ or of the form $\{x \in \Z_N : 1_A*1_A(x) < \gamma N\}$, where $A$ is a subset of $\Z_N$. One result we prove using this lemma is that if $|A| = \theta N$ and $|A+A| \leq (1-\eps) N$, $0 < \eps < 1$, then this level-set contains an arithmetic progression of length at least $N^c$, $c = c(\theta, \eps,\gamma) > 0$. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with some ideas of Bourgain); however, our method of proof allows us to obtain non-tower-type quantitative dependence between the constant $c$ and the parameters $\theta$ and $\eps$. For various reasons (discussed in the paper) one might think, wrongly, that such results would only be possible for level-sets involving triple and higher convolutions.